Chapter 2. Theoretical background

2.1 Vibronic coupling in organic molecules

Vibronic coupling is of great importance in spectroscopy. One can discern three main areas where vibronic coupling plays a major role. First, the absorption and emission of light are determined by the overlap of the vibrational wavefunctions of the initial and final states, as visualized in the vertical Franck-Condon principle. When the equilibrium geometries of the molecule in the states involved in transition are similar, i.e., small lateral displacements of the potential energy surfaces, and the force constants, i.e., the curvatures, are similar, the vibronic overlap between the two states is large and the transition is Franck-Condon allowed. Secondly, vibronic coupling determines the distortion introduced by degenerate electronic states. Thirdly, nonadiabatic radiationless transitions between states of different symmetry occur through vibronic coupling and are very important in photophysics.

The main aim of this paragraph is to present in the problem of vibronic coupling in polyatomic molecules in the presence of electronic degeneracy.

The idea of vibronic coupling and instability in a degenerate electronic state has first been formulated by Landau and Teller in 1934 (see for a reference [1]) and shown to be true for all nonlinear molecular systems by Jahn and Teller as formulated in the well-known Jahn-Teller theorem [2]. This effect is present in all situations where the electrons do not adiabatically follow the motion of the nuclei. Consequently, the nuclear positions are determined not only by the averaged field of the electrons, but also by the details of the electronic structure and their changes under nuclear displacements. This is always the case if electronic degeneracy is present.

2.1.1 The Born-Oppenheimer adiabatic representation

Consider an isolated molecule which has no interaction with any external field. The structure and the properties of this molecular system are determined by the motion of its electrons and nuclei and by their interaction. The main laws governing these motions have been discovered in the early thirties. However, because of mathematical difficulties, the quantum-mechanical treatment of molecular structures in most cases can be carried out only if some simplifying approximations (which are physically justified) are introduced. The most general of them is the Born-Oppenheimer adiabatic approximation which separates the time-scales of the electronic and nuclear motions.

The origin of the adiabatic approximation is the fact that the nuclear mass much larger than the mass of the electron (the nuclear mass of the H atom is 1837 times the electronic mass). Consequently, the electron velocity is much larger than that of the nucleus. Therefore, it can be assumed that every instantaneous (fixed) position of the nuclei corresponds to a stationary electronic state, and the motions of the nuclei are

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governed by the average field of the electrons. In other words, the nuclei are moving on a potential energy surface (PES) in the space of nuclear coordinates.

Under this assumption the Schrödinger equation can be solved in two stages [3]:

first, solving the electronic part of the problem by ignoring the nuclear motion,

secondly, using the mean electronic energy as the potential for the nuclear motion.

The total Hamiltonian that enters into the Schrödinger equation is:

(2.1)

where Hr is the electronic component including the kinetic energy, HQ is the kinetic energy of the nuclei, and V(r,Q) is the energy due to interaction of the electrons with the nuclei and the nuclear repulsion (r and Q are the whole set of coordinates of electrons ri and nuclei Q, respectively).

The operator V(r,Q) can be expanded as a series of small displacements of the nuclei about an origin point Q=Q0=0 :

(2.2)

V(r,0) is the potential energy of the electrons in the field of fixed nuclei.

One can solve the electronic part of Schrödinger equation (stage (i)):

(2.3)

and obtain a set of eigenvalues (energies)
and eigenfunctions (wave functions)
for a given nuclear configuration.

Now the question is how these solutions vary under nuclear displacements. Solving the full Schrödinger equation for different Q, one can obtain the PES‘s (stage (ii)).

The total Schrödinger equation is:

(2.4)

The total wavefunction is expanded in terms of electronic wavefunction:

(2.5)

Substituting equation (2.5) in equation (2.4) a set of coupled equations for the nuclear wave-functions
will be obtained:

(2.6)

where
is the electronic matrix element of vibronic interaction which depend on Q and is called vibronic coupling constant:

(2.7)

and
(2.8)

is the potential energy of the nuclei in the average field of the electrons in state
.

If second- and higher-order terms are neglected,
may be further explicitly written as:

(2.9)

For W(r,Q)=0 (i.e. no vibronic mixing) it is easy to see that the set of equations (2.6) transforms into a simple set of equations:

(2.10)

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which represents the Schrödinger equation for the nuclei moving in the mean field of the electrons in state
. In other words this is the adiabatic or Born-Oppenheimer approximation. It can be shown [3] that the perturbation of the total wave function by vibronic mixing is negligible if:

(2.11)

where the term on the left side is the energy quantum of vibrations in the electronic state under consideration (k or m) and
and
represents the energy levels m or k.

Equation (2.11) is a good criterion to discern whenever the adiabatic approximation can be applied. If this criterion is fulfilled, the error induced by the adiabatic approximation is of the order of (m/M)1/2 , where m and M are the electronic and nuclear masses, respectively.

2.1.2 Vibronic interactions and vibronic constants

Consider now a molecular system where two electronic states are degenerate or at least near-lying (pseudo-degeneracy). In this case, criterion (2.11) is not satisfied and vibronic mixing described by the operator (2.7) plays a significant role in describing the molecular properties.

To start the problem of vibronic coupling, Eq. (2.3) is solved for fixed nuclei, in order to determine the electronic levels and to find the electronic degeneracies. Then, Eq. (2.10) has to be solved in the normal coordinates of the molecule, to determine the frequencies of the normal modes. Finally, the matrix elements
of the coupling operator in Eq. (2.9) (called vibronic coupling constants) are calculated. They characterize the measure of coupling between the electronic structure and nuclear displacements, i.e. the measure of influence of the nuclear displacements on the electron distribution and, conversely, the effect of the changes in the electronic structure upon nuclear dynamics [3]. The first term in Eq. (2.7) is the linear vibronic constant and the second term is the quadratic vibronic constant, etc. In most cases, it is enough to take into account the linear and quadratic terms in order to reveal the vibronic effects. The linear vibronic constants have a clear physical meaning: they represent the force
(see Eq. (2.9)) with which the electrons affect the nuclei.

One of the effects of vibronic mixing is the appearance of anharmonicity in the potential energy surfaces of molecules, which has the origin in the existence of higher terms in the expansion (2.2).

2.1.3 Symmetry selection rules

The symmetry selection rule states that a transition or vibronic coupling is allowed only if the direct product of the representations involved in the process includes the fully

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symmetric species of the molecular point group:
. Here,
,
,
and
are the representations of the initial, transition (coupling), final and fully symmetric species.

The consequences that can be derived for Raman and electronic transitions, and for vibronic coupling will be discussed in the following.

Because the ground state wave function is usually fully symmetric, the Raman activity of a transition will be determined by the symmetry properties of the polarizability tensor components and of the vibrational wave function of the final state. If they belong to the same symmetry species, the full direct product of the transition will be symmetric and the transition is Raman active.

For an electronic transition, the direct product between the two electronic wave function and that of the transition dipole moment vector component should be totally symmetric. The ground state wave function is, as mentioned, totally symmetric. For molecules possessing a center of symmetry, it belongs to the gerade representation since it is not affected by inversion operation with respect to the center of symmetry. The dipole moment, however, changes its sign when the inversion operation is made. Consequently, it belongs to the ungerade representation. The electronic transition is thus allowed only if the excited state wave function belongs to the ungerade representation.

In the case of Diphenylhexatriene (n=3 in Fig. 1.1 right), which belongs to the C2h point symmetry group (two C2 axes and an inversion centrum), the ground and the first excited singlet state is of Ag symmetry and the second excited state of Bu symmetry. The Mulliken notation A and B for nondegenerate electronic states has been used [4].

This labeling scheme of the irreducible representations provides some additional information about their symmetry properties. All one-dimensional representations are labeled with A (for symmetric irreducible representations) or B (for antisymmetric irreducible representations). If a center of symmetry is present in a molecule, then g or u are used as a subscript to identify gerade (g) or ungerade (u) irreducible representations.

The consequence that can be derived from the symmetry selection rules in the case of Diphenylhexatriene is that only the transition from the ground state to the second excited singlet state (of Bu symmetry) is allowed.

The same selection rule applies for vibronic coupling between electronic states. The total vibronic and electronic symmetry must be examined. At this point it is important to mention that only nontotally symmetric displacements Qi can contribute to the matrix elements
, since the totally symmetric displacements do not change the symmetry and hence do not remove the degeneracy (considered as due to the symmetry of the system). Consequently, coupling between Ag and Bu electronic states in Diphenylhexatriene can only be made by nonsymmetric bu vibrations.

2.1.4 The Jahn-Teller theorem

Jahn and Teller proved that any nonlinear molecule in an orbitally degenerate electronic state will always distort in such a way as to lower the symmetry and remove the degeneracy [2].

The Jahn-Teller theorem is based on the analysis of the behavior of the adiabatic potentials of a polyatomic system near the point of electronic degeneracy. They showed [2, 4] that if the adiabatic potentials of a nonlinear polyatomic system have a n-fold degeneracy, at least one of them has no minimum at this point.

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Suppose now that by solving Eq. (2.3) for nuclei fixed at the point Qi=Qi0=0 a n-fold degeneracy is obtained with energies k‘=0. The question is how do these energy levels vary under nuclear displacements? In other words, how do the adiabatic potentials
look like for displaced coordinates with respect to the Qi=0 point? They can be obtained by estimating the effect of the vibronic interaction terms from Eq. (2.7) on the energy level positions k‘ determined with Eq. (2.3). Starting from the total Hamiltonian, the secular equation is solved:

(2.12)

where
are given by (2.7). Assuming small displacements, second- and higher-order terms in (2.7) may be omitted and
. Here
are the vibronic coupling constants, k and m are the degenerate electronic levels, and
are the modes along the displacements are made.

If at least one of
is nonzero, at least one of the roots
of Eq. (2.12) contains linear terms in the appropriate displacements
. Consequently, the adiabatic potential
has no minimum at the point
with respect to these displacements. This is illustrated in Fig. 2.1a for a two-fold degeneracy.

loses here the meaning of potential energy of the nuclei in the mean field of the electrons, since the motions of the electrons and nuclei near the point of degeneracy cannot be separated.

For linear molecules, the nontotally symmetric displacements are described as ungerade with respect to reflection, whereas the product of wave functions of the degenerate terms (coming from symmetry considerations) is always gerade with respect to the same symmetry operation. Consequently, all the linear
terms are zero, and only the quadratic terms in Eq. (2.7) could play a role. This effect is called Renner effect [6].

If the energy levels do not cross at Qi=0 but they are very closely spaced, the molecule is still unstable in the sense of Jahn and Teller, and the terminology used is pseudo-Jahn-Teller effect. This is illustrated in Fig. 2.1b.

The (pseudo) Jahn-Teller vibronic coupling effect can be described by a model Hamilton operator H. This operator takes the form of a nxn matrix in the case of n-fold degeneracy. In the simplest case of two-fold degeneracy, H becomes:

H=
(2.13)

where the diagonal elements correspond to the pure adiabatic states, and the non-diagonal elements give the coupling between the two states.

2.2 Time-dependent formalism for absorption and Raman scattering

In the precedent paragraph only stationary states of the molecule and their coupling were presented, without taking time-dependent phenomena into account.

For an analysis of the processes related to relaxation and dissipation of energy let us assume that a system S is excited with a very short pulse (see Fig. 2.2). After excitation, the system S will relax and dissipate the excitation energy to the environment E. The relaxation is here regarded as an internal process of energy redistribution along the internal degrees of freedom of the system. In contrast, dissipation is activated by the coupling to the external degrees of freedom and is associated with unidirectional flow of excess energy into the surroundings. In most cases, the environment E is a macroscopic system and will not increase its internal energy considerably.

Fig. 2.2: Schematic view of physical processes that can take place after excitation with an external field.

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The processes related to absorption, relaxation and dissipation have been treated within a variety of theoretical models. One of the most frequently used is the time-dependent semiclassical theory developed by Heller about twenty years ago [7], which neglect the dissipation processes.

The advantage of this theory is that it offers an intuitive picture of absorption, emission and Raman scattering in terms of nuclear wave packet movement on the potential surfaces. The wave packet dynamics corresponds to some specific vibrational motion that can be visualized like bond distances, angles, etc.

Let us take two adiabatic potential energy surfaces, each relevant for two vibrational degrees of freedom, x and y. Within the Born-Oppenheimer approximation one can write the vibrational wave function in the ground state
and vertically above it the wavefunction

(2.14)

with
the electronic transition moment between the two surfaces. The vertical transition is known as a Franck-Condon transition, and can be understood as an almost instantaneous electronic transition, while the nuclei retain their position and momentum. The transition can be induced by photoexcitation, for example. After the electrons have made a transition, the nuclei experience new forces. They find themselves displaced relative to the equilibrium geometry in the new potential surface and will be the subject of specific dynamics. In other words, the wave packet on the upper potential surface
is a displaced non-stationary one, which will evolve according to the time-dependent Schrödinger equation:

(2.15)

where
is the vibrational Hamiltonian for the upper surface

The absorption spectrum is:

(2.16)

where C is a constant,
is the frequency of the incident radiation, and E0 is the energy of
. In this way, the absorption spectrum is the Fourier transform of the overlap
. The overlap of the constant ground state wave function (at
) with the time dependent wave function in the excited state give rise to minima and maxima (they interfere constructively or destructively) specific for each vibrational motion. Usually, a very broad absorption spectrum is observed, due to the time-energy Uncertainty Principle where the broadest feature in the spectrum comes from the shortest feature in the time. For a downfall of the wave packet
with a time constant T1, the corresponding envelope will have the width
. Spreading and dissociation of the wave packet at later times contribute to its progressive amplitude lowering, which determine the substructure in the absorption spectrum.

In the Raman scattering process, the intensity is related to the square of the polarizability which can be defined in the time-dependent formalism like:

+(nonresonant term) (2.17)

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where
is the wave function of the final state n obtained by the multiplication of the wave function of the initial state with the transition moment µ and
is a damping factor which represents the effect of dephasing due to coupling to other degrees of freedom not explicitly included in the wave functions
. By comparing Eq. (2.17) with Eq. (2.16), it should be noted that the same dynamics of the same wave packet
is involved in both absorption and Raman scattering, but different final states
are involved.

2.3. A formalism for ultrafast electron transfer in condensed matter

Electron transfer (ET) is one of the most common reactions in both chemistry and biology, especially in the form of oxidation and reduction reactions. The ET process is, like any other chemical reaction, a transition from a metastable initial state to a stable final state. The initial state can be prepared in two different ways: photoabsorption or electron injection from external sources. The subsequent ET causes a redistribution of the electrostatic field in the molecule, which leads to a new equilibrium configuration of the nuclei. The interplay between the ET and the accompanying nuclear rearrangement is the key to understand the mechanism of ET. Because the ET reaction occurs in an environment which can play an active role in the reaction, the influence of the specific medium has to be taken into account.

The main aim of this chapter is to present a theoretical background of the ET theory from the classical Marcus theory to the quantum-mechanical theories describing the role of intramolecular vibrations in the ultrafast ET reactions.

2.2.1 Milestones in understanding of ET mechanism in solution

The first observation of ET reactions in solution goes back in the nineteenth century, when Humphry Davy observed (1808) that passing ammonia over metallic potassium produces a fine blue color. The modern experimental basis for ET reactions in solutions began 1920 with studies of ionic oxidation reduction reactions. The understanding of ET process has been considerably improved by Franck and Libby in 1949 [8]. They showed that ET transfer rates in solution are determined by horizontal Franck-Condon factors, in analogy to radiative processes where the transition probabilities are determined by vertical Franck-Condon factors. Between 1956-1960 R. A. Marcus made use of potential energy surfaces and statistical mechanics to provide a detailed classical description of the ET process [9-13]. He reduced the many-dimensional potential energy surfaces for reactants and products to harmonic free energy curves which were function of a single reaction coordinate. The so-called Marcus inverted regime was thus predicted (see the Paragraph 2.2.2) and later experimentally proved. The incorporation of quantum effects into the ET rate was advanced by Levich and Dogonadze [14, 15]. A theoretical description of the role of intramolecular vibrational excitations accompanying ultrafast ET reactions has been made by Jortner and Bixon [16, 17]. Currently, numerous groups of scientists are trying to understand the ET mechanism at a microscopic (vibrational) level and to explore the coherence phenomena related to the ET [18]. Another direction of recent interest is the possibility to control ET and to use molecules as molecular wire [18].

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2.2.2 The classical Marcus theory of ET

Consider a unimolecular ET reaction:

The transition occurs between the donor D and acceptor A assuming the horizontal Franck-Condon principle, i.e. the nuclear configuration of the reactant and product species must be the same at the point of transition state and the internal energy is conserved. The reactant D+A- has a potential energy which is a function of many nuclear coordinates (including solvent coordinates), resulting in a multidimensional potential energy surface. A similar surface has the product DA. In transition state theory, a reaction coordinate is introduced, so that the potential energy surface can be reduced to one- or a few-dimensional profile.

Marcus showed [9-13] that if the system is represented in a Gibbs (free) energy space, the Gibbs energy profiles along the reaction coordinate can be approximated as parabolas. The parabolic free-energy surfaces as a function of the reaction coordinate are illustrated in Fig. 2.3 for a variety of conditions. The curvatures of the reactant and product are assumed to be the same. The parameters shown in the diagram are: , the reorganization energy, which represents the change in free energy if the reactant were distorted to the equilibrium configuration of the product without transferring the electron; G0 which is the difference in free energy between the equilibrium configurations of the reactant and product states (the driving force) and G* which corresponds to the free energy of activation for ET. It is important to make the distinction between the G* and G+ which is the experimental free energy obtained from thermodynamic considerations in transition state theory.

From analytical geometry of intersecting parabolas it follows that:

(2.18)

According to classical transition state theory, the first-order rate constant
is given by:

(2.19)

where
is the electronic transmission coefficient which is related to the transition probability at the intersection of the two surfaces as determined by the Landau-Zener theory [19, 20] (
, for adiabatic and
for nonadiabatic ET, with Vel defined as the electronic coupling constant - see also the next paragraph),
is the frequency of nuclear passage through the transition state (typically 1013 s-1),
is the experimental free energy of activation,
the Boltzmann constant and
the temperature. Inserting (2.18) in (2.19), the classical Marcus equation is obtained:

(2.20)

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Eq. (2.20) indicates that for -G0 <, by increasing -G0 the ET rate kET increase. This is the normal regime (Fig. 2.3a). For -G0=, kET reaches its maximum
(Fig. 2.3b). For -G0 >, increasing further -G0 the ET rate kET decreases, and the inverted regime is obtained (Fig. 2.3c). The effect of decreasing kET in the inverted regime can be explained physically as follows: increasing the driving force -G0 to values larger than the reorganization energy leads to the increasing of the free energy of activation G*, i.e. barrier of the reaction.

Fig. 2.3: Three free energy regimes after Marcus (top) and the corresponding dependence of the transfer rates on G0 (bottom). By increasing the free energy of reaction -G0, the activation energy G* decreases leading to the increasing of the ET rate (normal regime). If -G0>, increasing further -G0 leads to the increasing of G* and consequently to the decreasing of the ET rate (inverted regime). R and P represent the reactant and product, respectively. RC is the reaction coordinate.

2.2.3 Inner-sphere versus outer-sphere ET

Polar solvents could influence the ET reactions. The change of the charge distribution taking place during the ET process is very sensitive to the permanent dipole moment

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carried by polar solvent molecules. If the ET is not too fast, the solvent molecules react by formation of a polarization cloud around the transfer complex, which follow the ET reaction and influences the ET rate. To characterize this influence, the macroscopic dielectric properties of the solvent comprised in the dielectric function
are used:

(2.21)

where
(
-refractive index of the medium) is the optical and
the static dielectric constant,
is the light frequency and
is the Debye dielectric relaxation time which is obtained by dielectric measurements.

If the ET is influenced mainly by solvent molecules, it is of outer-sphere type. The nuclear coordinates (and solvent degrees of freedom) which are coupled to the ET (i.e. are part of the reaction coordinate) are strongly perturbed by the solvent and the motion along the reaction coordinate becomes irregular, diffusion-like [21]. Consequently, the ET rate will be mainly determined by the inverse of the longitudinal dielectric relaxation time
. On the other hand, the ET is of inner-sphere type whenever intramolecular nuclear motions are dominant. In this case the motion along the reaction coordinate is only weakly perturbed by the solvent, and thus it is called uniform motion [21].

The main questions related to the outer-sphere ET can be summarized as follows:

(i) the competition between ET and solvent dielectric relaxation. If the solvent dielectric relaxation is very fast (typically less than 1 ps), the ET time is comparable to the solvent dielectric relaxation time. This is the solvent controlled regime. For slow dielectric relaxation (tens of picoseconds), the role of intramolecular nuclear motions increase and thus the ET time is less than the solvent dielectric relaxation time.

(ii) the interrelationship between the ET dynamics and the dissipative properties of the polar medium;

The main questions related in particular to the inner-sphere ET are:

(i) to discern between active and spectator modes in the ET

(ii) the interplay between vibrational excitation during ET and the consecutive vibrational relaxation.

The distinction between inner-sphere and outer-sphere ET is reflected in the reorganization energy
:

(2.22)

The solvent independent inner term
arises from structural differences between the equilibrium configurations of the reactant and product states. In the harmonic approximation, it can be written as:

(2.23)

where
is the reduced force constant for the i-th vibration,
are the equilibrium bond lengths in the reactant and product states, respectively, and the sum is taken over all active intramolecular vibrations which are part to the reaction coordinate.

The solvent dependent outer term
is called solvent reorganization energy and arises from differences between the orientation and polarization of solvent molecules around D+A-and DA. It represents the energy necessary to reorient the solvent molecules around the new equilibrium geometry of the product, but neglecting the additional

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effects due to ET. By treating the solvent as a dielectric continuum, the following expression can be derived for
[22]:

(2.24)

where B is a solvent-independent parameter depending on the model chosen and the molecular dimensions.
ranges from near zero for nonpolar solvents to 1.0 eV for polar solvents. It is also slightly temperature dependent, since both
and
vary with temperature. But for most liquids,
does not vary by more than 5% over a 100 K temperature range [22].

2.2.4 Adiabatic versus nonadiabatic ET

Two types of ET regimes can be distinguished according to the magnitude of the electronic coupling energy Vel between the reactant and product states, defined by:

(2.25)

where the
and
are the diabatic electronic wave functions of the equilibrium reactant and product states, respectively, and
is the electronic hamiltonian of the system calculated for rigid nuclei (Born-Oppenheimer approximation). For large Vel splitting, the ET reaction follows always the lower surface,
in Eq. (2.19) is equal to unity and the reaction is called adiabatic (Fig. 2.4 left). For very small Vel, the reactant and product surfaces do not interact significantly, the transition occurs only occasionally,
in Eq. (2.19) is close to zero and the reaction is called non-adiabatic (Fig. 2.4 right).

The physical meaning of this separation can be seen by introducing the typical times for electronic motion
and vibrational motion
, where
is the vibrational frequency [21].

If
(Vel large) , the electron will move many times between the donor D and acceptor A before any change in the nuclear configuration occurs. Consequently, this is the adiabatic case in the sense of Born-Oppenheimer.

Fig. 2.4: Adiabatic (left) and nonadiabatic (right) ET Vel is the same as defined in Eq. (2.25). The paths for ET are shown. P and R are the product and reactant diabatic states, respectively.

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If
(Vel small), the vibrational motion is much faster than the electronic one and the reaction is non-adiabatic.

2.2.5 Microscopic ET rates using the Fermi‘s Golden Rule

The classical Marcus theory presented in Paragraph 2.2.2 fits well with the experiment in the case of adiabatic ET reactions. For non-adiabatic reactions, however, nuclear tunneling between the reactant and product surfaces has to be taken into account.

Nuclear tunneling means that the stationary vibrational wave-functions is not strictly localized in the potential energy surface of reactant, but tails of it reach into the areas of the potential energy surface of the product. Vibrational overlap between the two surfaces becomes of major importance. The overlap between different vibrational wavefunctions in the reactant and product states is shown schematically in Fig. 2.5. One can see that in

Fig. 2.5: Schematic view of the vibrational overlap between the reactant R and product P vibrational wavefunctions in the normal (left) and in the Marcus inverted regime (right). v represents the vibrational quantum number.

the inverted regime nuclear tunneling plays a more important role than in the normal regime.

For the description of mode-selective ET, the concept of microscopic ET rate is introduced [18]. The transition rate constant ki from an initial vibrational level i in the reactant state to a set of vibrational levels f in the product state is given in the time-dependent perturbation theory in the form of the Golden Rule expression:

(2.26)

Here
and
are the vibrational wave functions for the equilibrium product f and reactant i states,
is the Dirac delta function ensuring the energy conservation,
and
are the vibrational energies of the i-th vibrational levels in the reactant and f-th vibrational level in the product states, respectively. Vel is the same as defined in Eq. (2.25) and the sum is taken over all internal and solvent vibrational modes in the

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product state.
are the Franck-Condon factors for levels i and f, representing the overlap between the two vibrational wavefunctions.

In some treatments [23] the solvent vibrations and the low frequency internal modes are considered classically, and thus the semiclassical Marcus equation can be written:

(2.27)

If the relevant high-frequency vibrations in the reactant state are replaced by one averaged mode with the frequency
, Eq. (2.27) transforms into [24]:

(2.28)

where n is an integer and

For a complete description of mode-selective ET, the mode-specific Franck-Condon factors should be expressed. In the simples approximation, the solvent contribution can be introduced in the form of line-shape function.